用机器学习优化卡拉比-丘度量，提升计算效率与精度。
Calabi-Yau metrics through Grassmannian learning and Donaldson's algorithm
- 在格拉斯曼流形上梯度下降，高效选取截面子空间。
- 在3维族上实现里奇平坦度量逼近，模参数增大时出现非平凡极小值。
- 结合唐纳森算法与矩阵学习，适合几何与物理交叉研究者。
受数值凯勒度量研究进展的启发，我们综述了该领域的机器学习技术，讨论其优劣。随后重新审视唐纳森开创的代数假设方法。受其启发，提出一种基于严谨框架的里奇平坦度量近似新方法，将机器学习应用于格拉斯曼流形上的梯度下降，以识别计算度量的有效截面子空间。该方法结合唐纳森算法与对$h$-矩阵的直接学习（等价于在格拉斯曼流形上正切丛的埃尔米特度量纤维丛上进行梯度下降）。我们在多尔克三重族上实现此方法，并分析了模空间不同点处的行为表现。特别地，随着模参数增加，观察到非平凡局部极小值的出现。
原文摘要 · Abstract (English)
Motivated by recent progress in the problem of numerical Kähler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired by his work, we present a novel approach to obtaining Ricci-flat approximations to Kähler metrics, applying machine learning within a `principled' framework. In particular, we use gradient descent on the Grassmannian manifold to identify an efficient subspace of sections for calculation of the metric. We combine this approach with both Donaldson's algorithm and learning on the $h$-matrix itself (the latter method being equivalent to gradient descent on the fibre bundle of Hermitian metrics on the tautological bundle over the Grassmannian). We implement our methods on the Dwork family of threefolds, commenting on the behaviour at different points in moduli space. In particular, we observe the emergence of nontrivial local minima as the moduli parameter is increased.
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